Matrix Diophantine equations over quadratic rings and their solutions

  • N.B. Ladzoryshyn Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, Ukraine
  • V.M. Petrychkovych Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, Ukraine
  • H.V. Zelisko Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine
Keywords: quadratic ring, matrix, $(z,k)$-equivalence of matrices, matrix Diophantine equation, solution of matrix equation
Published online: 2020-12-20

Abstract


The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In particular, solutions with bounded Euclidean norms are established. It is shown that there exists a finite number of such solutions of matrix equations over Euclidean imaginary quadratic rings. An effective method of constructing of such solutions is suggested.

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How to Cite
(1)
LadzoryshynN., PetrychkovychV., ZeliskoH. Matrix Diophantine Equations over Quadratic Rings and Their Solutions. Carpathian Math. Publ. 2020, 12 (2), 368-375.